This expression was first derived in the Lieb-Liniger model in [1]} using a property of relativistic quantum field theory called crossing symmetry, and the non-relativistic limit. Simple calculations then give the rather intuitive expression
\(\langle \hat{j}_n\rangle =\int {\rm d}p\, h_n(p)v^{\rm eff}(p)\rho _{\rm p}(p) \; ,\) 
